where subscript \(k\)  denotes the single particle state labeling, \(c_{k} = (-1)^{n_{1}+n_{2}+...+n_{k-1}} \in \lbrace  -1, +1\rbrace \)  and \(n_{1}, n_{2}, ...n_{k-1} \in \lbrace 0, 1\rbrace \)  are the particle number in each single particle state. \(X\) , \(Y\) , \(Z\)  are the Pauli gates [1]}. Nucleus contains both protons and neutrons and the transformation is carried for protons and neutrons separately. The operators should also be labeled with the kind of nucleon such as \(a^{\dagger }_{\pi , k}\) , \(a_{\pi , k} \) , \(a^{\dagger }_{\nu , k}\)  , \(a_{\nu , k} \) , \(I_{\pi }\)  and \(I_{\nu }\) .
